2
votes
1answer
38 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
0
votes
0answers
17 views

Milestein Scheme

Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ...
1
vote
0answers
29 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
1
vote
1answer
30 views

Evaluating Stratonovich integral from definition

I am struggling to evaluate the integral $\displaystyle \int^{T}_{0} B_t \circ dB_t $ from definition. So far I have that $\begin{align} \displaystyle \sum ...
0
votes
1answer
19 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
0
votes
0answers
26 views

Matlab code for Simulation of SDE [duplicate]

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
2
votes
2answers
104 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
0
votes
0answers
43 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
3
votes
0answers
66 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
0
votes
1answer
31 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
-3
votes
2answers
65 views

How to solve SDE

SDE: $dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T$ $X_0=a$ answer Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$ and ...
0
votes
0answers
20 views

Expected value of solution of SDE

Is there any way to find expectation of $X_t$ defined by the following SDE? $dX_t = -[\sin(2X(t)) + \frac{1}{4}\sin(4X(t))]dt + \sqrt{2}\cos^2 x dB(t), X(0)=1, t \in [0,\tau),$ where $\mathbb{B}$ is ...
0
votes
1answer
19 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
0
votes
1answer
28 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
0
votes
1answer
25 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
1
vote
1answer
94 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
0
votes
0answers
36 views

BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
0
votes
0answers
36 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
1
vote
1answer
28 views

I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
0
votes
0answers
33 views

Girsanov Measure Question.

If $Z_t = exp^{\int_0^t X_s dW_s - \frac{1}{2} \int_0^t (X_s)^2 ds}$ is a martinagle then by Girsanov's theorem, the measure $P_T$ defined by $P_T(A) = E^P(AZ_T)$ is mutually absolutely continuous ...
0
votes
1answer
49 views

Expectation of product of stochastic integral and brownian motion

Find the covariance: $$ COV((\int_t^T(T-s)dW_s), W_t) $$ I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ...
1
vote
1answer
41 views

Stochastic Integral Help

Let W(t) be a Brownian Motion. Show that the integral: $$ \int_t^T W(s)ds $$ can be written in terms of the stochastic integral: $$ \int_t^T (T-s)dW(S) $$ Is there an error with this question? I ...
0
votes
1answer
34 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
3
votes
1answer
27 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
0
votes
1answer
74 views

Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$ dV_t = -\beta V_t dt + \sigma dB_t $$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
1
vote
1answer
37 views

Poisson integral and discontinuous martingale (Ito-Levy formula)

Consider compounded Poisson process $P$ given by $P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz)$ where $N$ is a Poisson random measure of intensity $dt \otimes \nu$ and $\nu $ is a Levy measure. Why ...
1
vote
1answer
32 views

Elementary Malliavin Derivative question about definition.

I am reading a book that defines the Malliavin derivative $D_tF$ as follows: If $F = \sum_{n=0}^{\infty} I_n(f_n)$ is the Wiener Chaos expansion. $F$ is in the brownian filtration and $F \in ...
0
votes
1answer
64 views

Variance of Ito Integral

I want to find the variance of the Ito integral: $X(t)=\displaystyle \int_0^t\sqrt{s}WdW$ where W is a Brownian motion and s is the variable of integration. This is what I have done so far: ...
0
votes
1answer
54 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
2
votes
1answer
35 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
0
votes
1answer
36 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
1
vote
0answers
17 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
1
vote
1answer
38 views

Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
0
votes
0answers
15 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prevot/Rockner [PR07]: $ \int_0^T { \langle \Psi dW(t), \Phi(t)\rangle }$ A few useful ...
0
votes
0answers
25 views

Proposition from Oksendal Stochastic Calculus

I am reading Oksendal's Malliavin Calculus with applications to Finance and there is a part that I do not understand. First we have a proposition which is fine: If $\zeta_1$,$\zeta_2$,... are ...
0
votes
1answer
38 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
2
votes
0answers
51 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
1
vote
1answer
79 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
3
votes
1answer
83 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
4
votes
0answers
125 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
0
votes
0answers
128 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
1
vote
0answers
33 views

Proving $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$, $f$ a $k$-Lipschitz function, is a continuous martingale

Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function. I would like to show that $(\int_0^t f(X_s) ...
1
vote
1answer
40 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
0
votes
1answer
42 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
0
votes
1answer
51 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
0
votes
0answers
46 views

If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set?

Given a $X$ semi-martingale on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$ I am trying to prove: For any $B\in\mathcal F_\infty$ and processes $a_1,a_2$ such that ...
4
votes
0answers
91 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
0
votes
0answers
31 views

Differential of the integral of a stochastic process

In the HJM model one considers the forward rates to be on the form $$\mathrm df(t,T) = \alpha(t,T)\,\mathrm dt + \sigma(t,T)\,\mathrm dW(t)$$ In the proof of showing the drift condition on $\alpha$ ...
0
votes
1answer
42 views

Computation of a simple stochastic integral

For $t \in [0,T]$. consider two stochastic integrals with a nonnegative constant integrand $c$ $$\mathbb{E} \left[ \int_0^{t(\omega)^* \wedge T} c \cdot dW_t \right]$$ where $t^*$ is random ...
0
votes
1answer
41 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...